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Bader's Theory of Atoms in Molecules J. Chem. Sci. Vol. 128, No. 10, October 2016, pp. 1527–1536. c Indian Academy of Sciences. Special Issue on CHEMICAL BONDING DOI 10.1007/s12039-016-1172-3 PERSPECTIVE ARTICLE Bader’s Theory of Atoms in Molecules (AIM) and its Applications to Chemical Bonding P SHYAM VINOD KUMAR, V RAGHAVENDRA and V SUBRAMANIAN∗ Chemical Laboratory, CSIR-Central Leather Research Institute, Adyar, Chennai, Tamil Nadu 600 020, India e-mail: [email protected]; [email protected] MS received 22 July 2016; revised 17 August 2016; accepted 19 August 2016 Abstract. In this perspective article, the basic theory and applications of the “Quantum Theory of Atoms in Molecules” have been presented with examples from different categories of weak and hydrogen bonded molecular systems. Keywords. QTAIM; non-covalent interaction; chemical bonding; H-bonding; electron density 1. Introduction account of the bonding within a molecule or crystal. Specially, the derivative of electron density on BCP It is now possible to define the structure of molecules is zero. Bader’s group has made several seminal con- quantum mechanically with the help of Bader’s Quan- tributions to the development of the QTAIM theory tum Theory of Atoms in Molecules (QTAIM).1,2 This and its applications to unravel chemical bonding.10 13 theory has been widely applied to unravel atom-atom Popelier and coworkers have employed the QTAIM interactions in covalent and non-covalent interactions to address several issues in chemistry.14 16 Particu- in molecules,3 molecular clusters,4 small molecular larly, they have demonstrated the possibility of devel- crystals,5 proteins,6 DNA base pairing and stacking.7 oping structure-activity-relationship to predict various Basic motive of QTAIM is to exploit charge density or physico-chemical properties.17 They have used the the- electron density of molecules ρ(r; X) as a vehicle to ory of Quantum Chemical Topology (QCT), to provide study the nature of bonding in molecular systems. The ab initio descriptors that are able to accurately predict electron density (ED) in general can be defined as,2,8,9 pKa values for 228 carboxylic acids.18,19 Hydrogen- ∗ bond strengths have been described in terms of basici- ρ (r; X) = N dτ ψ (x; X)ψ(x; X) (1) ties with the help of several different scales. Bohorquez, Boyd, and Matta have illustrated the concept of deriving where N is the no. of electrons, x is the electronic coor- molecular model with quantum chemical bonding with dτ dinates, X is the nuclear coordinates and represents the help of the theory of QTAIM.1 Matta and cowork- the volume element of the system under consideration. ers have characterized the tri-hydrogen bond based on The ED distribution is predominantly affected by the topology of ED.20 interaction between two nuclei and hence the chem- Several groups have applied the QTAIM for unrav- ical bonding. It is found from previous reports that elling the non-covalent interactions such as weak van 4 5 both theoretical and experimental electron densities of der Waals, π...π,X-H...π, conventional hydrogen molecules can be used to gain insight into the nature of bonding, cation...π interactions, halogen bonds, etc. the chemical bonding. In fact, the topological analysis and also other applications in chemistry.3,21 24 All these of ED distributions has expanded beyond the territories interactions have been vividly characterized with the of theoretical chemistry to the X-ray crystallography. help of theory of QTAIM. It is not overstating that The topological properties of ED and its derivatives are QTAIM has completely provided new dimensions to the found to be immensely useful in delineating the con- concept of hydrogen bonding. Grabowski and cowork- cept of the bonding through bond paths and bond crit- ers have made several significant contributions to the 2,3 ical points (BCPs). The molecular graph and the ED characterization and understanding of hydrogen bond- along with its derivatives at the BCPs provide a precise ing interactions in various systems with the help of QTAIM.25,26 It is evident from previous studies that ∗For correspondence Bader’s topological parameters exhibited interrelation- Celebrating 100 years of Lewis Chemical Bond ship with geometrical parameters. For example, the 1527 1528 P Shyam Vinod Kumar et al. interrelationship between atom-atom distance and the is a linear relationship between interaction energy (IE) 3 ED at the BCP has been investigated. This relationship and ρ(rc) at the BCP for diverse class of molecular revealed that ED at the BCP provides a good measure systems.30 It is also evident from that report that weak, of strength of interaction between two molecular sys- moderate, strong and very strong hydrogen bonds can tems. It is also important to mention that QTAIM anal- be clearly described with the help of ED and its Lapla- ysis provided clear description of degree of covalency cian at BCPs. The relationship between Stabilization in the hydrogen bonding interaction. Espinosa and Energy (SE) and ED at BCP explain the smooth tran- coworkers27,28 have used parameters derived from the sition in the hydrogen strength from van der Waals to QTAIM analysis to elicit degree of covalency in hydro- covalent limit. gen bonding interaction. A measure of the strength The detailed theory of atoms in molecules has been 2a (com) of hydrogen bonding interaction has been devel- explained in the Bader’s Book. Other monographs oped with the help of QTAIM derived topological and on the QTAIM also provide basic theory and its geometrical parameters.29 It is given as: applications.31,32 In this perspective article, the basic theory and applications of QTAIM in eliciting the weak, ={[(r −r0 )/r 0 ]2 +[(ρ0 −ρ )/ρ 0 ]2 van der Waals, hydrogen bonding and ionic bonding com A−H A−H A−H A−H A−H A−H interactions in various molecules and molecular clus- 2 2 0 2 0 2 1 ters have been described with suitable examples. Since +[(∇ ρ − −∇ ρ )/∇ ρ ] } 2 (2) A H A−H A−H voluminous information is available on this topic, the comprehensive coverage of theory and applications of r ρ ∇2ρ where A−H, A−H and A−H represent proton donat- QTAIM is beyond the scope of this perspective article. ing bond involved in H-bonding, the bond length, elec- However, important aspects of the theory are provided tronic density at A-H bond critical point and Laplacian in this article. r0 ρ0 ∇2ρ0 of this density respectively. A−H, A−H and A−H correspond to the same parameters of the A-H bond 1.1 The basic theory of quantum theory of atoms involved in H-bond formation or any other nonbonded in molecule interaction. This complex parameter clearly describes the cova- It is well known that the ED distribution of a molecular lency of hydrogen bonding interaction. Higher values of system is the physical manifestation of the forces act- 2 com indicate lengthening of the proton donating bond ing within the molecular system. The attractive force and shorter proton-acceptor distances. Higher value of exerted by the nuclei determines the most important com implies greater binding energy and negative value topological properties of ED of multi-electronic sys- of the total electron energy density at hydrogen bond tem. The ED exhibits local maxima only at the posi- critical point(HBCP) and hence the degree of cova- tions of the nuclei. Exceptions to this have also been lency of hydrogen bonding interaction. In addition to observed under certain circumstances. It is possible to QTAIM, electron delocalization function (ELF), Nat- recognize atomic forms within the molecules with the ural Bond Orbital analysis (NBO) and energy decom- help of local maxima exhibited by the ED at the nuclei. position analysis (EDA) are also useful in describing The ED distribution, ED relief map and molecular the covalency of hydrogen bonding interaction.2 In graph of benzene are shown in Figure 1, which clearly understanding degree of covalency of hydrogen bond- reveal the presence of local maxima at the position ing interaction, Parthasarthi et al., have shown that there of nucleus. Figure 1. (a) Electron density contour map; (b) Relief map; (c) Molecular graph of Benzene. Atoms in Molecules and Chemical Bonding 1529 1.2 Topography of electron density distribution its associated basin.2 It can be seen from Figure 1c and critical points that (3, −1) CP is found between every pair of nuclei which are linked by a chemical bond in the benzene It can be inferred from Figure 1 that the ED exhibits molecule. It is found from detailed analysis of topolog- a maximum, a minimum, or a saddle point in space. ical features of ED that pairs of gradient paths which These special points are referred to as Critical Points originate at each (3, −1) CP and terminate at the neigh- ρ( ) (CPs). At this point, the first derivatives of rc van- boring attractors.2 These two unique paths describe a ∇ρ( ) = ∇ρ( ) ishes, i.e., rc 0, where rc is given in line through the charge distribution connecting neigh- equation (3) and rc is the CP. boring nuclei along which ρ is a maximum with respect ∂ρ ∂ρ ∂ρ to any neighboring line. Such a line is known as ∇ρ(r ) = i + j + k c ∂x ∂y ∂z (3) atomic interaction line in the topological analysis of ED distribution.2 The presence of a (3, −1) CP and It is well known that a maximum or a minimum or related atomic interaction line highlights that electronic an extremum is determined by the sign of its second charge density is accumulated between the nuclei that derivative at this point.
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